Central and non-central limit theorems in a free probability setting

TL;DR

This paper explores non-central limit theorems within free probability, demonstrating that certain free probability time series converge to limits involving non-commutative processes like free fractional Brownian motion.

Contribution

It introduces free probability analogs of classical long-range dependence results, replacing Hermite processes with non-commutative Tchebycheff processes and analyzing their limit behaviors.

Findings

Limits are represented by multiple Wigner integrals.

Non-commutative fractional Brownian motion is characterized.

Non-commutative Rosenblatt process is analyzed.

Abstract

Long-range dependence in time series may yield non-central limit theorems. We show that there are analogous time series in free probability with limits represented by multiple Wigner integrals, where Hermite processes are replaced by non-commutative Tchebycheff processes. This includes the non-commutative fractional Brownian motion and the non-commutative Rosenblatt process.